Comparison with exact solutions

As the semiclassical tunneling theory and Bardeen s original approach become inaccurate for potential barriers close to or lower than the energy level, the validity range of the MBA is much wider. In this subsection, the accuracy of the MBA is tested against an exactly soluble case, that is, the one-dimensional transmission through a. square barrier of thickness W=2 A (see Fig. 2.9). 

The exact transmission coefficient is presented in Equations Eq. (2.8) and (2.10). The logarithm of the exact transmission coefficient varies almost linearly with barrier height, all the way from 4 eV above the energy level to 2 eV below the energy level. 

The transmission coefficient of the MBA can be evaluated analytically. For the case of L = W2, the result is 

As shown in Fig. 2.10, the transmission coefficient of the MBA remains accurate even with a barrier 2 eV below the energy level. By taking L = WI3, a similar result is found, as shown in Fig. 2.9. Therefore, the accuracy of MBA is approximately independent of the choice of the separation surface. The transmission coefficient of the original BA can be obtained by letting 

Ko = K in Eq. (2.44). As shown in Fig. 2.9, the BA becomes inaccurate when the barrier top is close to or lower than the energy level. The accuracy of semiclassical theory is even worse than the BA, as shown in Fig. 2.10. As the top of the barrier comes close to the energy level, the WKB transmission coefficient quickly becomes unity. The MBA with corrections with Green s function is also shown. For the case of Fig. 2.9, the values of the integrals are ( - UbL) and [ - Uk W -L) respectively. As shown, the result is fairly accurate. 

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Calculated results by DLT and FLT in comparison with exact solution. 

Comparison with exact solutions of the Poisson-Boltzmann equation showed that expression (4.63) gives adequate agreement for low surface potentials [418]. For eX g-x gq (4.63) can be simplified ... 

One way to examine the validity of the steady-state approximation is to compare concentration—time curves calculated with exact solutions and with steady-state solutions. Figure 3-10 shows such a comparison for Scheme XIV and the parameters, ki = 0.01 s , k i = 1 s , 2 = 2 s . The period during which the concentration of the intermediate builds up from its initial value of zero to the quasi-steady-state when dcfjdt is vei small is called the pre-steady-state or transient stage in Fig. 3-10 this lasts for about 2 s. For the remainder of the reaction (over 500 s) the steady-state and exact solutions are in excellent agreement. Because the concen-... 

The purpose of this chapter is to provide a comprehensive discussion of some simple approaches that can be employed to obtain information on the rate of heat and mass transfer for both laminar and turbulent motion. One approach is based on dimensional scaling and hence ignores the transport equations. Another, while based on the transport equations, does not solve them in the conventional way. Instead, it replaces them by some algebraic expressions, which are obtained by what could be called physical scaling. The constants involved in these expressions are determined by comparison with exact asymptotic solutions. Finally, the turbulent motion is represented as a succession of simple laminar motions. The characteristic length and velocity scales of these laminar motions are determined by dimensional scaling. It is instructive to begin the presentation with an outline of the basic ideas. 

Examples from quantum mechanics. The examples hitherto considered have been chosen with a view to illustrating various aspects and defficulties of p.m. in comparison mith exact solutions. In consequence it was inevitable that they were rather of artificial character except for Ex. 2, which is, however, an example in which p.m. is not valid. So it would be desirable to discuss some examples taken from problems of quantum mechanics. 

In Secs. II.A and II.B above, we examined some common, approximate solutions to the linear Poisson-Boltzmann equation, and commented on the level of their agreement with exact solutions of that same equation. However, these approximations are no more accurate than the exact solutions, and the accuracy of the latter can only be ascertained by comparison with solutions to the complete, nonlinear Poisson-Boltzmann equation. From the... 

One way to examine the validity of the steady-state approximation is to compare concentration-time curves calculated with exact solutions and with steady-state solutions. Figure 3-10 shows such a comparison for Scheme XIV and the parameters, ki = 0.01 s , k i = 1 s , = 2 s . The period during which the concentration... 

Comparison with Exact Results. It is not unreasonable to suspect that truncation errors in the numerical approximation of first and second derivatives might accumulate in the computational scheme used to integrate the mass transfer equation. One check for accuracy involves a comparison between numerical results and exact analytical solutions. Of course, only a limited number of analytical solutions are available. For example, the following solutions have been obtained analytically for catalytic duct reactors ... 

In this section, NOVA-3D predictions are verified against the exact solution for non-Unear creep and recovery of an IM7/5260 [90]i6 specimen for two different stress levels. Figure 12.6 shows a comparison of the creep and recovery of an IM7/5260 [90]ig specimen predicted by NOVA-3D with exact solution. The constitutive law expressed in equation [12.12] was used for NOVA-3D predictions. The exact solution for this case is given by ... 

Figure 12.6 Comparison of NOVA-3D predictions with exact solution for transverse creep and recovery of an IM7/5260 [901,5 specimen. ----, NOVA-3D +, exact 21 MPa O, exact 70MPa...

Let us start the illustration with the situation where < = 1, that is, the rate of reaction is comparable to the rate of diffusion. Using the techniques taught in Chapters 2 and 3, the exact solution to Eqs. 8.25 (which is needed later as a basis for comparison with approximate solutions) is... 

Figure 9. Comparison of measured solids residence time with exact solution of Kramers and Croockewit [45] (for all configurations) O, without lifters and no end constriction , with lifters and no end constriction O, without lifters and with end constriction A, with lifters and with end constriction , Abouzeid and Fuerstenau [11,29].

The error in figure 3.3 and 3.4 has been specified by comparison with a solution with SFD, 20 000 discretization points and LIMEX, RTOL=10 . It is easily seen that the CFD performs much better for a given number of discretization points than the SFD. The overall error is very small and the maximum temperature is met almost exactly. In figure 3.4 the cpu-time needed on a Sun Sparestation 2 to reach a certain accuracy is compared. In addition to the SFD and CFD, orthogonal collocation on finite elements (OCFE) with 2 collocation points per finite element [12] is included. It is evident, that the higher order methods CFD and OCFE perform much better than SFD. 

Cooney, D.O., Comparison of simple adsorber breakthrough curve method with exact solution, AIChE J., 39(2), 355-358 (1993). 

The plot of (2.157) with the parameters from Table 2.1 is shown in Figure 2.15. For comparison the exact solution to Eq. (2.155) is shown (dashed line both curves are practically indistinguishable). As seen, temperature variation across the CL is less than 0.1 K and it can safely be ignored. However, the temperature gradient at i = 1 is not small. This gradient (given by Eq. (2.149)) determines the heat flux from the CL to the GDL. 

Comparison of Euler s method with exact solution... 

FIG. 3.5. Comparison of Jander equations with exact solutions. 

Figure 13.31. Comparison of numerical solution from Figure 13.30 with exact solution values as given by Eq. (13.57).

We shall examine the simplest possible molecular orbital problem, calculation of the bond energy and bond length of the hydrogen molecule ion Hj. Although of no practical significance, is of theoretical importance because the complete quantum mechanical calculation of its bond energy can be canied out by both exact and approximate methods. This pemiits comparison of the exact quantum mechanical solution with the solution obtained by various approximate techniques so that a judgment can be made as to the efficacy of the approximate methods. Exact quantum mechanical calculations cannot be carried out on more complicated molecular systems, hence the importance of the one exact molecular solution we do have. We wish to have a three-way comparison i) exact theoretical, ii) experimental, and iii) approximate theoretical. 

Thus, the error from ignoring the bend-twist coupiing terms is about 24%, certainly not a negligible error. Hence, the specially orthotropic laminated plate is an unacceptable approximation to a symmetric angle-ply laminated plate. Recognize, however, that Ashton s Rayleigh-Ritz results are also approximate because only a finite number of terms were used in the deflection approximation. Thus, a comparison of his results with an exact solution would lend more confidence to the rejection of the specially orthotropic laminated plate approximation. 

Figure 3-10. Comparison of steady-state and exact solutions for the concentration of intermediate B (relative to Ca) for Scheme XIV, where ki = O.OI s , k t = I s 2 = 2 s . The exact. solution was obtained with Eq. (3-87), the. steady-.state solution with Eq. (3-142), where Ca was calculated with Eq. (3-90).

In Eq. (3.66) the sign + is chosen to provide the decay in time of the spectrum correlation function. When the approximate solution (3.66) is used for the back iterations in Eq. (3.58) from bN = 1 + bN up to ho and subsequent calculation of ao(co) the error does not accumulate. This was proved by comparison of approximate numerical calculations of limiting cases 2 and 3 with exact formulae (3.61) and (3.62). 

First, we have applied the ZN formulas to the DHj system to confirm that the method works well in comparison with the exact quantum mechanical numerical solutions [50]. Importance of the classically forbidden transitions has been clearly demonstrated. The LZ formula gives a bit too small results... 

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